Talk Abstract The application of the x-ray transform to 3D conformal radiotherapy
with dynamic multileaf collimators

Robert Y. Levine, MIT

It is shown that for $\gamma$-ray $(>1 MeV)$ therapy the delivered
dose can be approximated by the dual attenuated x-ray transform
of the filtered beam profiles. The implied treatment geometry
is appropriate for the new multileaf collimators. The number
of intensity-modulated beams required for conformal radiotherapy
is examined using the mathematics of tomographic reconstruction.
For a 2D tomotherapy geometry the sampling requirement is at
most $(2 \pi r_{max} W_{max} + 5/2)$ beams, where $r_{max}$
and $W_{max}$ are the maximum spatial extent and frequency,
respectively, of the radiation dose. We generalize this ``Bow
Tie" solution to 3D, suggesting a sufficient beam number given
by $(\Delta \omega / 2 \pi W_{max})(2 \pi r_{max} W_{max} +
5/2)^{2}$, where $\Delta \omega$ is the frequency resolution
of the beam front modulation delivered by the multilear collimator.
The matrix inversion implicit in this bound suggests a beam
selection criteria. The beam angles should be chosen such that
the SVD inversion to beam profiles is non-singular for the entire
configuration. The direct function metric among beam profiles
provides another criteria for choosing beam angles. By maximizing
the overlap between the sampled and continuous beam profile
functions, the intensity (in the $\rho$-metric) is derived and
displayed relative to the 3D data set for a ranking of beam
orientations.

The formalism above is applied to the derivation and evaluation
of radiotherapy plans for brain and prostate tumors based on
real patient data from the UMASS Medical School and Massachusetts
General Hospital. Dose-volume histograms are examined for evidence
of beam number thresholds in conformal treatments.